For the design and manufacturing companies in various industries, where the fabrication of their products needs to warp 2D sheet material (e.g., metal in ship industry, fabric in apparel industry and toy industry, and leather in shoe industry and furniture industry) into a 3D shape without any a stretching, it becomes a bottleneck to determine a shape of 2D pieces according to a designed 3D surface patch in the design and manufacturing cycle now. The problem of surface flattening (or parametrization) is usually formulated under a constrained optimization framework, the resultant 3D patch generally is not a developable surface and its length is not invariant. For an engineering application like a 3D garment design and manufacturing, this length variation will lead to many problems if pieces are sewed together, and the garment shape and designed fit will be affected.
In industries whose products are fabricated from planar pieces of sheet material, designers desire a surface flattening tool, which can preserve the length of boundaries and feature curves on a 2D piece according to its 3D surface patch. Although there are many different related approaches existing towards 3D modeling or surface flattening for pattern design, they are either time consuming or do not have the property of length preservation.
For example, there are some researches in literature which propose some approaches to focus on modeling or approximating a model with developable ruled surfaces (or ruled surfaces in other representations—e.g. B-spline or Bézier patches). However, these approaches can only model surface patches with 4-sided boundaries and it is difficult to use these approaches to model freeform surfaces as the surfaces are usually not defined on a square parametric domain. Although trimmed surfaces can be considered, the modeling ability for freeform objects by this category of approaches is still very limited. In short, the proposed approaches can only design the objects with relative simple shapes.
An ideal surface flattening of a given 3D surface patch P to be flattened to its corresponding 2D flattened piece D preserves the distances between any two points. That is, an isometric mapping is needed mathematically. However, this property is only held on those developable surfaces. Therefore, the existing surface flattening approaches always evaluate the error of distance variations between surface points on P and D, and try to minimize this error under a non-linear optimization framework. Unfortunately, the computation of non-linear optimization in terms of vertex position is very time-consuming and can hardly preserve the invariant lengths of feature curves.
Another interesting category of surface flattening approaches solves the problem by computing mappings for dimensionality reduction or through a multidimensional scaling (MDS) technique. These approaches are all based on computing an optimal mapping that projects the geodesic distances on surfaces into Euclidean distances in R2 (i.e. for a lower dimension space). Nevertheless, it is difficult to embed the hard constraints on the length of feature curves in the mapping computation.
For length preservation of feature curves, J. R. Manning suggested the idea of preserving the length of feature curves on a network in “Computerized pattern cutting: methods based on an isometric tree”, Computer-Aided Design, Vol. 12, No. 1, PP. 43-47, 1980, wherein an isometric tree consisting of a network of curves that are mapped onto the plane isometrically is introduced. However, this network is with the tree topology and the isometric curves are the branches of the tree which are flattened one by one without considering the relationship between these curves.
In the document “Piecewise surface flattening for non-distorted texture mapping” published in Computer Graphics, Vol. 24, No. 4, PP. 237-246, 1991, Bennis et al. mapped isoparametric curves onto plane followed by a relaxation process to position the surface between them and employed a progressive algorithm to process complex surfaces; however, the relationship between these isoparametric curves was not well addressed.
In the document “Geodesic curvature preservation in surface flattening through constrained global optimization”, Computer-Aided Design, Vol. 33, No. 8, PP. 581-591, 2001, Azariadis and Aspragathos also proposed a method for optimal geodesic curvature preservation in surface flattening with feature curves. Nevertheless, as it was based on an optimization in terms of vertex positions, it is highly nonlinear and cannot be efficiently solved.
In literature, some approaches directly model developable (or flattenable) surfaces in R3 instead of computing a surface flattening mapping. As proposed in the document “Virtual garments: a fully geometric approach for clothing design”, Computer Graphics Forum (Eurographics'06 Proceedings), Vol. 25, No. 3, PP. 625-634. 2006, a given mesh surface is processed by fitting a conical surface locally at every vertex so that expected normal vectors can be determined. More generally, the discrete definition of Gaussian curvature has been adopted to define the measurement for the developability on given polygonal mesh surfaces in “Achieving developability of a polygonal surface by minimum deformation: a study of global and local optimization approaches”, The Visual Computer, Vol. 20, No. 8-9, PP. 521-539, 1052, where a constrained optimization approach was conducted to deform mesh surfaces to increase their discrete developability. Liu et al. presented a novel PQ mesh in “Geometric modeling with conical meshes and developable surfaces”, ACM Transactions on Graphics, Vol. 25, No. 3, PP. 681-689, 2006, which can be used to model developable surfaces in strips. Recently, a FL mesh modeling scheme which models developable mesh surfaces with a more complicated shape has been presented. However, it is never easy to modify any of these approaches so that they can process a surface from non-developable to developable while preserving the length of feature curves. Besides, the computation is much slower.
Therefore, a fast surface flattening approach which can warp a given 3D surface into 2D with the lengths of edges of its boundaries and feature curves being preserved is in a great need.